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(2020) Machine Learning Holographic Mapping by Neural Network PHYSICAL REVIEW RESEARCH 2, 023369 (2020) Machine learning holographic mapping by neural network renormalization group Hong-Ye Hu ,1 Shuo-Hui Li ,2,3 Lei Wang,2,4,5 and Yi-Zhuang You 1,* 1Department of Physics, University of California at San Diego, La Jolla, California 92093, USA 2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3University of Chinese Academy of Sciences, Beijing 100049, China 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Received 28 February 2020; accepted 1 June 2020; published 19 June 2020) Exact holographic mapping (EHM) provides an explicit duality map between a conformal field theory (CFT) configuration and a massive field propagating on an emergent classical geometry. However, designing the optimal holographic mapping is challenging. Here we introduce the neural network renormalization group as a universal approach to design generic EHM for interacting field theories. Given a field theory action, we train a flow-based hierarchical deep generative neural network to reproduce the boundary field ensemble from uncorrelated bulk field fluctuations. In this way, the neural network develops the optimal renormalization-group transformations. Using the machine-designed EHM to map the CFT back to a bulk effective action, we determine the bulk geodesic distance from the residual mutual information. We have shown that the geometry measured in this way is the classical saddle-point geometry. We apply this approach to the complex φ4 theory in two-dimensional Euclidian space-time in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry when geometric fluctuation is neglected. DOI: 10.1103/PhysRevResearch.2.023369 I. INTRODUCTION theory of different scales are mapped to different depths in the bulk, and vice versa. The field variable deep in the The holographic duality, also known as the anti-de Sitter bulk represents the overall or infrared (IR) features, while space and conformal field theory correspondence (AdS-CFT) the variable close to the boundary controls the detailed or [1–4], is a duality between a CFT on a flat boundary and a ultraviolet (UV) features. Such a hierarchical arrangement gravitational theory in the AdS bulk with one higher dimen- of information is often observed in deep neural networks, sion. It is intrinsically related to the renormalization-group particularly in convolutional neural networks (CNNs) [16]. (RG) flow [5–11] of the boundary quantum field theory, since The similarity between renormalization group and deep learn- the dilation transformation, as a part of the conformal group, ing has been discussed in several works [17–22]. With the naturally corresponds to the coarse-graining procedure in the development of machine learning in quantum many-body RG flow. The extra dimension emergent in the holographic physics [23–28], deep learning techniques have also been bulk can be interpreted as the RG scale. In the traditional employed to construct the optimal RG transformations [29,30] real-space RG [12], the coarse-graining procedure decimates and to uncover the holographic geometry [31–34]. In this irrelevant degrees of freedom along the RG flow, therefore work, we further explore the possibility of designing the EHM the RG transformation is irreversible due to information loss. for interacting quantum field theories using deep learning However, if the decimated degrees of freedom are collected approaches. By training a flow-based hierarchical generative and hosted in the bulk, the RG transformation becomes a model [35,36] to generate field configurations following the bijective map between the degrees of freedom on the CFT probability distribution specified by the field theory action, the boundary and the degrees of freedom in the AdS bulk. model converges to the optimal EHM with an emergent bulk Such mappings, generated by information-preserving RG gravitational description, where neural network parameters transforms, are called exact holographic mappings (EHMs) and latent variables correspond respectively to the gravity [13–15], which were first formulated for free-fermion CFT. (geometry) and matter degrees of freedom in the holographic Under the EHM, the boundary features of a quantum field bulk. The learned holographic mapping is useful for both sam- pling and inference tasks. In the sampling task, the generative model is used to propose efficient global updates for boundary *[email protected] field configurations, which helps to boost the Monte Carlo simulation efficiency of the CFT. In the inference task, the Published by the American Physical Society under the terms of the boundary field theory is pulled back to an effective theory of Creative Commons Attribution 4.0 International license. Further the bulk field, which enables us to probe the emergent dual distribution of this work must maintain attribution to the author(s) geometry (on the classical level) by measuring the mutual and the published article’s title, journal citation, and DOI. information in the bulk field. 2643-1564/2020/2(2)/023369(12) 023369-1 Published by the American Physical Society HU, LI, WANG, AND YOU PHYSICAL REVIEW RESEARCH 2, 023369 (2020) field on the holographic boundary. The inverse RG can be considered as a deep generative model G, which organizes the bulk field ζ (x, z) to generate the boundary field φ(x), φ(x) = G[ζ (x, z)]. (1) The renormalization G−1 and generation G procedures are thus unified as the forward and backward maps of a bijective (invertible) map between the boundary and the bulk, known as the EHM [13,14]. At first glance, such an information-preserving RG does not seem to have much practical use, because ζ (x, z)issimply a rewriting of φ(x), which does not seem to reduce the degrees of freedom. However, since the bulk field ζ (x, z) represents the irrelevant feature to be decimated under RG, it should look like independent random noise, which contains a minimal amount of information. So instead of memorizing the bulk FIG. 1. Relation between (a) RG and (b) generative model. The field configuration ζ (x, z) at each RG scale for reconstruction inverse RG can be viewed as a generative model that generates the ζ , ensemble of field configurations from random sources. The random purposes, we can simply sample (x z) from uncorrelated sources are supplied at different RG scales (coordinated by z), which (or weakly correlated) random source and serve them to the ζ , can be viewed as a field ζ (x, z) living in the holographic bulk with inverse RG transformation. Suppose the bulk field (x z)is ζ one more dimension. The original field φ(x) is generated on the drawn from a prior distribution Pprior[ ], the transformation holographic boundary. φ = G[ζ ] will deform the prior distribution to a posterior distribution Ppost[φ] for the boundary field φ(x), II. RENORMALIZATION GROUP − δG[ζ ] 1 AND GENERATIVE MODEL P φ = P ζ , post[ ] prior[ ]det δζ (2) Renormalization group (RG) plays a central role in the −1 study of quantum field theory (QFT) and many-body physics. where | det(δζ G)| is the Jacobian determinant of transfor- The RG transformation progressively coarse-grains the field mation. In such a manner, the objective of the inverse RG configuration to extract relevant features. The coarse-graining is not to reconstruct a particular original field configuration, rules (or the RG schemes) are generally model dependent but to generate an ensemble of field configurations φ(x), φ and require human design. Take the real-space RG [12]for whose probability distribution Ppost[ ] should better match the example: for a ferromagnetic Ising model, the RG rule should Boltzmann distribution, progressively extract the uniform spin components as the −S [φ] P [φ] = e QFT /Z , (3) most relevant feature; however, for an antiferromagnetic Ising target QFT φ model, the staggered spin components should be extracted specified by the action functional SQFT[ (x)] of the boundary − φ = SQFT[ ] instead; if the spin couplings are randomly distributed on field theory, where ZQFT [φ] e denotes the partition the lattice, the RG rule can be more complicated. When it function. comes to the momentum-space RG [37], the rule becomes This setup provides us a theoretical framework to discuss to renormalize the low-energy degrees of freedom by inte- the designing principles of a good RG scheme. We propose grating out the high-energy degrees of freedom. What is the two objectives for a good RG scheme (or EHM): the RG general design principle behind all these seemly different RG transformation should aim at decimating irrelevant features schemes? Can a machine learn to design the optimal RG and preserving relevant features, and the inverse RG must scheme based on the model action? aim at generating field configurations matching the target field With these questions in mind, we take a closer look at theory distribution Ptarget[φ]inEq.(3). An information theo- the RG procedure in a lattice field theory setting. In the retic criterion for “irrelevant” features is that they should have traditional RG approach, the RG transformation is invertible minimal mutual information, so the prior distribution Pprior[ζ ] due to information loss at each RG step when the irrelevant should be chosen to minimize the mutual information between features are decimated, as illustrated in Fig. 1(a). However, bulk fields at different points, i.e., min I(ζ (x, z):ζ (x, z )). if the decimated features are kept at each RG scale, the RG We refer to this designing principle as the minimal bulk transformation can be inverted. Under the inverse RG flow, mutual information (minBMI) principle, which is a general the decimated degrees of freedom ζ (x, z) are supplied to information theoretic principle behind different RG schemes each layer (step) of the inverse RG transformation such that and is independent of the notion of field pattern or energy the field configuration φ(x) can be regenerated, as shown in scale.
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